Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [calculus-of-variations]

Optimization of functionals mostly defined on infinite-dimensional spaces.

0
votes
0answers
18 views

Find the extremals of the functional $\int\sqrt{x^2+y^2}\sqrt{1+(y'(x))^2}dx$?

I want to use the polar coordinates $x=r\cos\theta$ and $y=r\sin\theta$. After transformation, I get $$\int r\sqrt{r^2+(r')^2}d\theta.$$ Then, I derived the Euler-Lagrange equation $$2r^2-rr''+3(r')^2=...
0
votes
0answers
18 views

Guaranteeing isoperimetry constraint for non-extremal functional in PDE.

First of all, hello and thank you for your time. Context I am making a program that solves the differential equation for the time evolution of a system from the equations: $$F[\mathbf{y}]=\int\...
1
vote
2answers
79 views

Solving the Euler-Lagrange equation for the brachistochrone problem with friction

This Wolfram Alpha Page contains a derivation of the parametric form of the brachistochrone curve that result from either assuming friction or its absence. I am asking for help understanding how ...
2
votes
1answer
70 views

Why is compactness important to find solution of certain problems?

I've been recently re-thinking about my knowledge of compacts sets and I've realized that other than the definition and maybe understanding with some struggle some proofs I don't actually see why they'...
1
vote
0answers
26 views

Derivatives of the eikonal

Let $y$ be an extremal function of the functional $$J[y]=\int_{x_1}^{x_2} L(x, y(x), y'(x)) \mathrm{d}x$$ So it satisfies the Euler-Lagrange equation: $$\frac{\partial L}{\partial y}-\frac{\mathrm{d}}{...
0
votes
2answers
54 views

Geometrical explanation

I would like to describe you a technique that I found in a physics paper, for which I cannot give a mathematical interpretation. Let $f(q,g,e,r)=\frac{r}{2}(g+q) + q^{\frac{p}{2}} e - \frac{T}{2} \...
1
vote
1answer
63 views

Existence of a minimizer by means of direct methodes

The following is given: Let $\Omega\subset \mathbb{R}^n$ be a bounded, connected open set with Lipschitz boundary. Let $f\in C^0(\overline{\Omega}\times \mathbb{R}\times \mathbb{R}^n)$, $f=f(x,u,\xi)$,...
1
vote
1answer
92 views

proving an interpolation inequality in $L^p$ norms

Let $1\le p \le \infty$. Prove that for all $\epsilon >0$ there exists a constant $C>0$ such that $$\|u'\|_{L^p(\mathbb{R})}\le \epsilon \|u''\|_{L^p(\mathbb{R})}+C\|u\|_{L^p(\mathbb{R})} \;\; \...
3
votes
0answers
21 views

How to determine if an equation is an Euler-Lagrange equation of some functional?

Given an equation (e.g., a PDE), is there any way to determine if it is an Euler-Lagrange equation of some functional? If yes, is there in general any method to find out the functional? Or must there ...
0
votes
1answer
26 views

How are the steps to the solution for Arc - Length obtained?

Can someone please help me follow and understand the steps of the solution marked with $(*)$ and $(@)$? Why is the dot product used and computed with the unit vector. How does this equal the integral? ...
0
votes
0answers
30 views

How to minimize the next functional using the Pontryagin Maximum Principle?

Best regards, I am asked to minimize the next functional $T(v)=\int_{A}^{B}\frac{dx}{v(x)}$, with $v\neq 0\text{ and } v\in V$, where $V=\{v\in C([A,B]):v(A)=v(B)=0\}$. If we assume that there ...
0
votes
1answer
32 views

Point to curve problem: How to solve the Euler's equation?

I'm trying to solve the following problem: Find the extremal curve for $ J(y) = \int_0^{x_1} \frac{ ( 1 + (y')^2)^{\frac{1}{2}}}{y} dx = \int_0^{x_1} F(y,y') dx $ with $y(0) = 0$ and the point $(x_1, ...
1
vote
1answer
50 views

With the solution of the Euler-Lagrange equation prove the following equation

Let $\Omega \subset \mathbb{R}^n$ and $F=F(z,p):\Omega \times \mathbb{R} \times \mathbb{R}^n$ be smooth and independent of $x\in\Omega$. Let $u$ be a solution of the Euler-Lagrange equation of $\...
3
votes
1answer
44 views

Asymptotic behaviour of gradient flows for $t \to \infty$

I have often heard about the asymptotics of gradient flows converging to some "equilibrium point" as $t \to \infty$. This concept has come to my ear by word of mouth multiple times and is often ...
4
votes
0answers
80 views

A standard example of calculus of variation that I don't understand

Let us consider the problem $$ \min \left\{ \int_0^1 \psi\left(\dot{u}\right) \ dx: u \in \mathcal{C}^1([0,1]), u(0) =0, u(1)=2 \right\} $$ where $\psi \colon \mathbb{R} \longrightarrow \mathbb{R}$...
0
votes
1answer
31 views

Euler-Lagrange equation for the Brachistochrone problem with friction

The WolframMathworld website has a page containing the derivation for the brachistochrone with and without friction. I am able to follow the derivation for the with-friction case up to the point of ...
0
votes
0answers
19 views

Why is the distributional derivative of a continuous function non-atomic?

For $\Omega \subset \mathbb{R}^N$ and $u \in BV(\Omega)$, we define the distributional derivative $Du$ as the N-tuple of finite Radon measures in $\Omega$, $Du = (D_1u,...,D_Nu)$, such that $$ \int_{\...
0
votes
0answers
18 views

Does functional satisfies Palais-Smale condition?

Check if the functional $f(u)=\int_0^{1/2} u^2(x)dx$ satisfies the Palais-Smale condition on the Hilbert space $L^2([0,1],\mathbb{R})$. We have definied the Palais-Smale condition as follows: $f$ ...
4
votes
1answer
44 views

Second Lemma of variation Calculus to prove the Euler-Lagrange equation

Second Lemma of variation Calculus If $u(x)$ is a differentiable function for $a\leq x \leq b$ and $$ \int_{a}^{b} u(x) \cdot \phi'(x)\: dx = 0$$ for all infinitely often differentiable functions $\...
2
votes
0answers
22 views

Criterion for weakly lower semicontinuity in problems of calculus of variations

Let’s say I have to minimize a certain integral functional $F(u)= \int_{\Omega} L(x,u,\nabla u)$ with $\Omega$ regular and $u$ in a certain Sobolev space (don’t really care to be precise, I just want ...
0
votes
1answer
29 views

Euler-Lagrange equation in polar or cylindrical coordinates

I have been studying Euler-Lagrange in Variation Calculus. I am comfortable with the formulation when the function under the integral is of the form f = f(x, y). ...
3
votes
1answer
37 views

Converting ODE to Variational Problem (for numerical solution)

This might be a stupid question, but I cannot find the answer anywhere and as an engineer I don't have the mathematical foundation to investigate this properly myself. So, If I have a simple ODE, say ...
3
votes
1answer
43 views

Do variations obey the product rule?

I have been trying to derive the Einstein equation from the Einstein-Hilbert action $$ S[g_{\mu \nu}] = \frac{1}{16 \pi} \int_M \text{d}^4x \sqrt{-g}R $$ The standard derivation states that the ...
-1
votes
0answers
21 views

Exercise about Sobolev Spaces

I'm studying a course about Sobolev spaces and differential equations but i'm struggling to find a book where to find good exercises about these arguments. Do someone know a book like this?
3
votes
0answers
63 views

How to treat an equation of the form $-\Delta u=G\cdot \nabla u+f(u) ?$

There are plenty of variational techniques (direct methods of calculus of variations, mountain pass type theorems, Lusternik-Schnirelmann theory) to prove the existence of solutions of a semilinear ...
2
votes
2answers
65 views

Variation of a metric $g$ with signature (1,-1,-1,-1)

I'm new in variations of metric. Let be $g$ a metric with signature (1,-1,-1,-1) on a manifold. If I consider a family of variations $g+\varepsilon h$ (depending on $h$), used to derive the Eulero-...
0
votes
1answer
32 views

A problem in variational calculus

How do you maximize the quotient ||f||/||f'|| of euclidean norms if f is to be a function on [0,1] which vanishes on the boundary? $||g||^2 = \int_0^1g(x)^2\textrm{d}x$ I guess $f$ needs to be ...
1
vote
1answer
78 views

“Standard methods of the calculus of variations” or “Do you read German?”

I'm trying to understand an article of Reinsch (1967), on smoothing spline functions. The author uses some rules that unfortunately I couldn't find. The minimized functional is: $$ \int_{x_0}^{x_n} g'...
0
votes
0answers
46 views

Conditional extremum with Lagrange multipliers

I have the following problem : Find the extremals of the functional : where lampda is Lagrange multipliers my question is how to solve the differential equation below sorry L dont =0
0
votes
0answers
42 views

Sturm-Liouville eigenvalues have lower bound

I am looking to find a proof that eigenvalues are bounded below for the the general Sturm Liouville equation on an open region $ \Omega \subset \mathfrak R^d $ (perhaps with compact closure), $Ly -\...
1
vote
0answers
25 views

Young measure generated by sequence

It's about the following exercise: $i)$ Let $h: \mathbb{R} \rightarrow \mathbb{R}$ with $h(x)=\begin{cases} a,\; 0 \leq x < \lambda \\ b, \; \lambda \leq x < 1.\end{cases}$. The function $h$ ...
1
vote
2answers
44 views

Minimizer of square root operator norm

Let $A:D(A) \to \mathcal H$ be a positive self-adjoint operator and $\sqrt{A}$ defined by via the spectral theorem on $D(\sqrt{A}) = Q(A)$ where $Q(A)$ is the quadratic form domain. Let $$E=\inf\{\...
0
votes
0answers
16 views

Using Euler's equation to calculate integral of $\zeta$ over the curve $Re(\zeta(t)=0)$ between first two zeros?

Is Euler's equation the right equation to use if I want to calculate a numerical value of $\int_{\{ {Re} (\zeta (t)) = 0 : 14.134 \ldots < {Im} (t) < 21.022 \ldots . \}} \zeta (t) d t$ It ...
0
votes
1answer
28 views

a question related to calculus of variations

Consider a particle with coordinates $(x(t),y(t))$ on a smooth curve $\phi(x,y)=0$. If the particle moves from $(x(0),y(0))$ to $(x(\tau),y(\tau))$ for $\tau >0$ such that its kinetic energy is ...
4
votes
2answers
66 views

Minimize $\int_0^1 A(x)dx$ given $\int_0^1 \frac{dx}{A(x)}\le \text{constant}$

The following problem is derived from a Finite element problem. We have a beam which is subject tot a constant axial force. With out loss of generality the beam can have a variable section area $A(x)$...
1
vote
1answer
42 views

Exercise using the derivation of the Euler-Lagrange equation

Here is a exercice using the derivation of the Euler-Lagrange equation: Here is the exercice: For a given function $f(x,u,u')$ and constants $K_1, K_2$ minimize the functional using the Euler-...
0
votes
0answers
37 views

Existence of minimizers in Evans chapter 8

I am trying to understand the following point in Evans PDE book. In chapter 8, page 465, he proved the existence of minimizer to the following problem, let $U$ be open, bounded with connected, $C^1$ ...
0
votes
0answers
54 views

Formulation for calculus of variation with state-space constraint

I'm stuck on this question, let $B = \{x\in \mathbb{R}^n:|x|\leq 1\}$ be the unit ball in $\mathbb{R}^n$, consider the following minimizing problem $$ \inf_{x(\cdot) \in \mathcal{A}} \int_0^\infty e^{-...
2
votes
1answer
33 views

Lower semi continuous envelope is lower semi continuous

Let X be a topological space and $F:X \rightarrow \overline{\mathbb{R}}$. The lower semi continuous envelope of $F$ is defined by $sc^-F(u)=\sup\{\phi(u)\;|\; \phi :X \rightarrow \overline{\mathbb{R}} ...
1
vote
1answer
52 views

Does a surface with given boundary in $\mathbb{R}^3$ exist?

Given a (smooth) simple closed curve $C \subset \mathbb{R}^3$, is there a (smooth) surface $S$ with $\partial S = C$? I'm aware there is a variational problem to find among such surfaces the one with ...
-2
votes
1answer
32 views

Finding functional extremals

So my problem is as follows: find the extremals of the functional $$I[x_1(t), x_2(t)]=\int_{0.5}^1(\dot{x}_1^{2}-2x_{1}\dot{x}_2t)dt,$$ given: $$x_1(0.5)=2, \ \ \ \ x_2(0.5)=15, \ \ \ \ x_1(1)=1, \...
1
vote
1answer
63 views

Prove that $u=v$

I got the following integral identity $$\int_{\Omega}\left[H(\nabla u)(\nabla H)(\nabla u)-H(\nabla v)(\nabla H)(\nabla v)\right]\cdot\nabla\left(u-v\right)\;dx=0$$ and i want to prove that $u=v$. ...
2
votes
0answers
46 views

Distributional second-order derivatives of $\frac{e^{-|x|}}{4\pi |x|}$ to show the solution of $u -\Delta u=f$ is in $H^2$

In Brezis's book "Functional Anlaysis" it is proven that the solutions of the Helmotz equation $u - \Delta u=f$ where $f \in L^2 (\mathbb{\Omega})$ belong to $H^2 (\mathbb{\Omega}) \cap H^1 _0 (\Omega)...
4
votes
1answer
85 views

Finding Christoffel Symbols using via variational method.

I'm trying to find the Christoffel Symbols for the Lorentz metric $${\rm d}s^2 = \cos(2\pi x)({\rm d}x^2-{\rm d}y^2) - 2\sin(2\pi x)\,{\rm d}x\,{\rm d}y$$by looking at the Euler-Lagrange equations for ...
1
vote
1answer
37 views

How to take derivative of integral of function?

I'm reading a textbook where it forms a Lagrangian function $$ L = \int_0^1 f(x)^{1 - \frac{1}{\alpha}}dx - \lambda\int_0^1 g(x) f(x) dx$$ But how do you take the derivative of this thing? The ...
2
votes
0answers
79 views

Optimization in Banach space: Find functions that minimize the supremum of a convex operator.

Let $D \subset \mathbb{R}^n$ be compact. Denote by $C(D, \mathbb{R}^n)$ the space of continuous functions from $D$ to $\mathbb{R}^n$. Let $K$ be a real, symmetric, positive-definite $n \times n$ ...
1
vote
0answers
20 views

Intuition behind Pohozaev identity

I'm wondering if the Pohozaev identity: $$n\int_\Omega\int_0^{u(x)}f(t)\operatorname{d}t\operatorname{d}x-\frac{n-2}{2}\int_\Omega u(x)f(u(x))\operatorname{d}x=\frac{1}{2}\int_{\partial\Omega}\left|\...
0
votes
0answers
117 views

Numerical Methods for Euler-Lagrange Equations

For my coursework I've been told to write an algorithm in Python to solve the Euler-Lagrange equations with Dirichlet boundary conditions. This is my first Numerical Methods module. We have briefly ...
1
vote
0answers
14 views

rate-distortion function: derivative of a functional

My question concerns the answer on this post: Characterisation of the rate distortion function: issue with functional derivative What do the symbols $\delta_{x',x_0}$ and $\delta_{\hat{x}_0,\hat{x}}$ ...
4
votes
0answers
88 views

Does this functional satisfies the Palais-Smale condition?

Let $\Omega$ be a non-empty bounded open subset of $\mathbb{R}^N$, $\lambda\in \mathbb{R}$ be an eigenvalue of $-\Delta$ on the Sobolev space $H^1_0(\Omega)$ and $f\in L^\infty(\Omega\times\mathbb{R})$...